Computer methods and algorithms can be used to analyze and solve complex systems involving various forms of fluid dynamics having inputted boundary conditions. For example, computer modelling may allow a user to simulate the flow of air and other gases over an object or model the flow of fluid through a pipe. Computational fluid dynamics (CFD) is often used with high-speed computers to simulate the interaction of one or more fluids over a surface of an object defined by certain boundary conditions. Typical methods involve large systems of equations and complex computer modelling and include traditional finite difference methodology, cell-centered finite volume methodology and vertex-centered finite volume methodology.
Traditional Finite Difference Methodology
Traditional Finite Difference Methodology (TFDM) requires a structured grid system, a rectangular domain and uniformed grid spacing. TFDM cannot be applied on a mesh system with triangular cells (elements). Rather, cells must be quadrilateral (2D) and cannot be polygonal (i.e., number of sides=4). In 3D, cells must be rectangular cubes.
TFDM typically requires the use of coordinate transformations (i.e., grid generation) for curvilinear domains, to map the physical domain to a suitable computational domain. In addition, there may be a need to use a multiblock scheme if the physical domain is too complicated. Partial differential equations (PDEs) must be transformed to the computational domain.
Traditional Finite Difference Methodology is typically difficult to deal with in complicated grid arrangements. Special treatment may be required near boundaries of the domain (e.g., in staggered grid systems or for higher-order schemes). Even with coordinate transformations, highly irregular domains may create serious difficulties for accuracy and convergence due to numerical discontinuities in the transformation metrics. Cell-Centered Finite Volume Methodology/Vertex-Centered Finite Volume Methodology
Cell-Centered Finite Volume Methodology (CCFVM) and Vertex-Centered Finite
Volume Methodology (VCFVM) achieve greater flexibility in grid arrangement. Cells can be polygonal (e.g., triangular) in 2 Dimensional space or polyhedral (e.g., tetrahedral, prismatic) in 3 Dimensional space. With CCFVM/VCFVM there is no need for coordinate transformations to a computational domain. Rather, all calculations can be done in physical space. As well, grid smoothness is not an issue. Cell-centered schemes evaluate the dependent variable at the centroid of each cell. Vertex-centered (or vertex-based) schemes evaluate the dependent variable at the vertices of each cell.
With CCFVM/VCFVM, inaccuracies due to calculation of fluxes across cell faces may be difficult to deal with. In addition, there are difficulties associated with treatment near boundaries for higher-order schemes, and accuracy and convergence issues associated with cells that are severely skewed or have a high aspect ratio.
Accordingly, current computer modelling schemes are limited in the form of objects they can model and require different models and algorithms for different fluid applications, such as between compressible and non-compressible fluids.